Second-order lower radial tangent derivatives and applications to set-valued optimization

Journal of Inequalities and Applications, Jan 2017

We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated. MSC: 46G05, 90C29, 90C46.

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Second-order lower radial tangent derivatives and applications to set-valued optimization

Xu et al. Journal of Inequalities and Applications Second-order lower radial tangent derivatives and applications to set-valued optimization Bihang Xu 2 Zhenhua Peng 0 1 Yihong Xu 0 0 Department of Mathematics, Nanchang University , Nanchang, 330031 , China 1 School of Mathematics and Statistics, Wuhan University , Wuhan, 430072 , China 2 School of Information Engineering, Nanchang University , Nanchang, 330031 , China We introduce the concepts of second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for set-valued maps by means of a radial tangent cone, second-order radial tangent set, lower radial tangent cone, and second-order lower radial tangent set, respectively. Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated. Henig efficiency; radial tangent derivative; set-valued optimization; optimality condition 1 Introduction © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. gent derivatives of the objective function and the constraint function are not separated, and thus the properties of the derivatives of the objective function are not easily obtained from those of the constraint function. On the other hand, some efficient points exhibit certain abnormal properties. To eliminate such anomalous efficient points, various concepts of proper efficiency have been introduced [–]. Henig [] introduced the concept of Henig efficiency, which is very important for the study of set-valued optimization [, , , ]. In this paper, we introduce a new class of lower radial tangent cones and two new kinds of second-order tangent sets, using which we introduce four new kinds of second-order tangent derivatives. We discuss the properties of these second-order tangent derivatives, using which we establish second-order necessary optimality conditions for a point pair to be a Henig efficient element of a set-valued optimization problem. 2 Basic concepts Throughout the paper, let X, Y , and Z be three real normed linear spaces, X , Y , and Z denote the original points of X, Y , and Z, respectively. Let M be a nonempty subset of Y . As usual, we denote the interior, closure, and cone hull of M by int M, cl M, and cone M, respectively. The cone hull of M is defined by dom F := x ∈ X : F(x) = ∅ , graph F := (x, y) ∈ X × Y : y ∈ F(x) , epi F := (x, y) ∈ X × Y : y ∈ F(x) + C . Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A. The radial tangent cone of A at xˆ, denoted by R(A, xˆ), is given by Remark . Equation (.) is equivalent to where N denotes the set of positive integers. Definition . (See []) Let A be a nonempty subset of X, and let xˆ ∈ cl A. The contingent cone of A at xˆ, denoted by T (A, xˆ), is given by Remark . (See []) Equation (.) is equivalent to   T (A, xˆ, w) := v ∈ X : ∃tn → + and vn → v such that xˆ + tnw +  tnvn ∈ A . Definition . (See [, ]) Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈ X × Y . The second-order composed contingent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the set-valued map D F(xˆ, yˆ, uˆ , vˆ) : X → Y defined by Definition . (See []) Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈ X × Y . The second-order contingent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the set-valued map DF(xˆ, yˆ, uˆ , vˆ) : X → Y defined by In the following, we introduce a new class of lower radial tangent cones and two new kinds of second-order tangent sets. Definition . Let Q be a nonempty subset of X × Y , and let (xˆ, yˆ) ∈ cl Q. The lower radial tangent cone of Q at (xˆ, yˆ) is defined by such that (xˆ + tnun, yˆ + tnvn) ∈ Q . Definition . Let Q be a nonempty subset of X × Y , and let (xˆ, yˆ) ∈ cl Q. The secondorder lower radial tangent set of Q at (xˆ, yˆ) in the direction (uˆ , vˆ), denoted by Rl(Q, (xˆ, yˆ), (uˆ , vˆ)), is given by Definition . Let A be a nonempty subset of X, and let xˆ ∈ cl A. The second-order radial tangent set of A at xˆ in the direction w, denoted by R(A, xˆ, w), is given by   R(A, xˆ, w) := v ∈ X : ∃tn >  and vn → v such that xˆ + tnw +  tnvn ∈ A . Example . Let R be the set of real numbers, X = Y = R, Q = {(– n , n ) : n = , , . . .} ∪ {(x, y) : x ≥ , y ≥ } ∪ {(–, –)}, and (xˆ, yˆ) = (uˆ , vˆ) = (, ). A direct calculation gives Rl(Q, (, ), (, )) = {(x, y) : x > , y ≥ }, T (Q, (, ), (, )) = {(x, y) : x ≥ , y ≥ } ∪ {(x, ) : x < }, and R(Q, (, ), (, )) = {(x, y) : x ≥ , y ≥ } ∪ {(x, ) : x < } ∪ {(x, x) : x < } ∪ n∞={λ(– n , n ) : λ > }. 3 The second-order lower radial tangent derivative In this section, by virtue of the radial tangent cone, the second-order radial tangent set, the lower radial tangent cone, and the second-order lower radial tangent set, we introduce the concepts of the second-order radial composed tangent derivative, the second-order radial tangent derivative, the second-order lower radial composed tangent derivative, and the second-order lower radial tangent derivative for a set-valued map. Furthermore, we discuss some important properties of the second-order lower radial composed tangent derivative and the second-order lower radial tangent derivative. Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F , and (uˆ , vˆ) ∈ X × Y . The second-order radial composed tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the set-valued map R F(xˆ, yˆ, uˆ , vˆ) : X → Y defined by If R(R(epi F, (xˆ, yˆ)), (uˆ , vˆ)) = ∅, then F is said to be second-order radial composed derivable at (xˆ, yˆ) in the direction (uˆ , vˆ) or that the second-order radial composed tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) exists. Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F , and (uˆ , vˆ) ∈ X × Y . The second-order radial tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the setvalued map RF(xˆ, yˆ, uˆ , vˆ) : X → Y defined by If R(epi F, (xˆ, yˆ), (uˆ , vˆ)) = ∅, then F is called second-order radial derivable at (xˆ, yˆ) in the direction (uˆ , vˆ) or that the second-order radial tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) exists. Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈ X × Y . The second-order lower radial composed tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the set-valued map Rl F(xˆ, yˆ, uˆ , vˆ) : X → Y defined by If Rl(Rl(epi F, (xˆ, yˆ)), (uˆ , vˆ)) = ∅, then F is said to be second-order lower radial composed derivable at (xˆ, yˆ) in the direction (uˆ , vˆ) or that the second-order lower radial composed tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) exists. Definition . Let F : X → Y be a set-valued map, (xˆ, yˆ) ∈ graph F, and (uˆ , vˆ) ∈ X × Y . The second-order lower radial tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) is the set-valued map RlF(xˆ, yˆ, uˆ , vˆ) : X → Y defined by If Rl(epi F, (xˆ, yˆ), (uˆ , vˆ)) = ∅, then F is called second-order lower radial derivable at (xˆ, yˆ) in the direction (uˆ , vˆ) or that the second-order lower radial tangent derivative of F at (xˆ, yˆ) in the direction (uˆ , vˆ) exists. Proposition . Suppose that E ⊂ X and the second-order lower radial composed tangent derivative of F : X → Y at (xˆ, yˆ) ∈ graph F in the direction (uˆ , vˆ) exists. Then Rl F(xˆ, yˆ, uˆ , vˆ) R R(E, xˆ), uˆ ⊂ clcone clcone F(E) + C – yˆ – vˆ . v ∈ Rl F(xˆ, yˆ, uˆ , vˆ)(u). uˆ + tnun ∈ R(E, xˆ). xˆ + tnkukn ∈ E. Therefore, there exist sequences tnk >  and ukn → uˆ + tnun such that (uˆ + tnun, vˆ + tnvn) ∈ Rl epi F, (xˆ, yˆ) . xˆ + tnkukn, yˆ + tnkvkn ∈ epi F, Then, for the same tnk and ukn, there exists a sequence vkn → vˆ + tnvn such that and, consequently, v + t v as k → ˆ n n → ∞ , we obtain n n ∈ n ∈ and, consequently, → ∞ , we get n ∈ cone clcone clcone clcone and, consequently, R F(x, y, u, v) R (E, x, u) ˆ ˆ ˆ ˆ ˆ ˆ l clcone cone F(E) + C – y – v . ˆ ˆ Proposition . Suppose that E X and the second-order lower radial tangent derivative of F : X F in the direction (u, v) exists. Then ˆ ˆ Proof Let v   R F(x, y, u, v)(R (E, x, u)). Then there exists u ˆ ˆ ˆ ˆ ˆ ˆ l R F(x, y, u, v) R R(E, x), u ˆ ˆ ˆ ˆ ˆ ˆ l clcone clcone F(E) + C – y – v . ˆ ˆ R F(x, y, u, v) = R ˆ ˆ ˆ ˆ l l u such that For such t and u , it follows from (.) that there exists a sequence v n n v such that and, consequently, n ∈ n ∈ n n ∈ n n ∈ n ∈ cone cone → ∞ , we get clcone cone   R F(x, y, u, v) R (E, x, u) ˆ ˆ ˆ ˆ l ˆ ˆ ⊂ clcone cone F(E) + C – y – v . ˆ ˆ D F(x, y, u, v) R R(E, x), u ˆ ˆ ˆ ˆ ˆ ˆ ⊂ clcone clcone R F(x, y, u, v) R R(E, x), u ˆ ˆ ˆ ˆ ˆ ˆ ⊂ clcone clcone Remark . If we substitute D F(x, y, u, v) or R F(x, y, u, v) for R F(x, y, u, v) in Proposiˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ l tion ., then none of the inclusions is necessarily true. If we substitute DF(xˆ, yˆ, uˆ , vˆ) or RF(xˆ, yˆ, uˆ , vˆ) for RlF(xˆ, yˆ, uˆ, vˆ) in Proposition ., then none of the inclusions is necessarily true, as is shown in the following example. Example . Let R be the set of real numbers, X = Y = R, C = {t : t ≥ }, and E = {x : x ≥ }. Define the set-valued map F : X → Y by F(x) = {y : y ≥ } if x ≥ , √ {y : y ≥  x} otherwise. R(E, ) = R R(E, ),  = [, +∞), D F(, , , –)(x) = R F(, , , –)(x) = Rl epi F, (, ) = (x, y) : x ∈ R, y ≥  , Rl Rl epi F, (, ) , (, –) = ∅, Rl F(, , , –)(x) = ∅, x ∈ R. D F(, , , –) R R(E, ),  = R F(, , , –) R R(E, ),  = R, Rl F(, , , –) R R(E, ),  = ∅, Then, the inclusion of Proposition . is true. However, D F(xˆ, yˆ, uˆ , vˆ) R R(E, xˆ), uˆ ⊂ clcone clcone F(E) + C – yˆ – vˆ D F(, , , )(x) = R F(, , , )(x) = R R F, (, ), (, ) ≥ } ≥ } R F(, , , )(x) = R F(, , , )(x) = [, + ), x ∞ l l F, (, ) = (x, y) : x = R = T F, (, ), (, ) F, (, ) = R = (x, y) : x > , y (x, y) : x R F(x, y, u, v) R R(E, x), u ˆ ˆ ˆ ˆ ˆ ˆ clcone clcone F(E) + C – y – v . ˆ ˆ (ii) Let (x, y) = (, ), (u, v) = (, ). A direct calculation gives ˆ ˆ ˆ ˆ R R(E, ),  = R (E, , ) = R(E, ) = [, + ), ∞ F, (, ) , (, ) = R R F, (, ) , (, ) = T F, (, ), (, ) D F(, , , ) R R(E, ),  = R F(, , , ) R R(E, ),  = D F(, , , ) R (E, , ) = R F(, , , ) R (E, , ) = R, R F(, , , ) R R(E, ),  l = R F(, , , ) R (E, , ) = [, + ), ∞ l clcone clcone F(E) + C – y – v = ˆ ˆ clcone cone F(E) + C – y – v = [, + ). ˆ ˆ ∞ Then, the inclusions of Propositions . and . are true. However, D F(x, y, u, v) R R(E, x), u ˆ ˆ ˆ ˆ ˆ ˆ R F(x, y, u, v) R R(E, x), u ˆ ˆ ˆ ˆ ˆ ˆ D F(x, y, u, v) R (E, x, u) ˆ ˆ ˆ ˆ ˆ ˆ clcone clcone clcone clcone clcone cone F(E) + C – y – v , ˆ ˆ F(E) + C – y – v , ˆ ˆ F(E) + C – y – v , ˆ ˆ R F(x, y, u, v) R (E, x, u) ˆ ˆ ˆ ˆ ˆ ˆ clcone cone F(E) + C – y – v . ˆ ˆ 4 Second-order necessary optimality conditions Let F : X Z  , and (F, G) : X be defined by (F, G)(x) = F(x) Consider the following optimization problem with set-valued maps: min F(x), s.t. G(x) ∩ (–D) = ∅, x ∈ X. Definition . (See [, , ]) Let xˆ ∈ Eˆ , yˆ ∈ F(xˆ). A pair (xˆ, yˆ) is called a Henig efficient element of (VP) if there exists ε ∈ (, δ) such that where δ := inf{ b : b ∈ B}, F(Eˆ ) = Definition . (See []) The interior tangent cone IT(S, y¯) of S at y¯ is the set of all y ∈ Y such that for any tn → + and yn → y, we have y¯ + tnyn ∈ S. IT(S, y¯) = IT(int S, y¯) = intcone(S – y¯). Theorem . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D), (uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), F is second-order lower radial composed derivable at (xˆ, yˆ) in the direction (uˆ , vˆ), and G is second-order radial composed derivable at (xˆ, zˆ) in the direction (uˆ , wˆ ). Then there exists εˆ ∈ (, δ) such that On the contrary, suppose that (.) does not hold. Then there exist x¯ ∈ dom Rl F(xˆ, yˆ, uˆ , vˆ) ∩ dom R G(xˆ, zˆ, uˆ , wˆ ), y¯ ∈ Rl F(xˆ, yˆ, uˆ , vˆ)(x¯), and z¯ ∈ R G(xˆ, zˆ, uˆ, wˆ )(x¯) such that z¯ ∈ – int D. Since (u , w ) R( n n ∈ k k k G, (x, z)), there exist sequences t >  and (x , z ) ˆ ˆ n n n G such that Hence, there exist t >  and (u , w ) R( n n n ∈ G, (x, z)) such that ˆ ˆ t (u , w ) – (u, w) n n n ˆ ˆ t (w – w) – n n ˆ ∈ D is a cone, we obtain From (.) it follows that there exists N  ∈ N such that –D and –D is a convex cone, it follows that n ∈ D – D = – N such that n > N , k > K (n).  ∀  D is a cone, we obtain n > N , k > K (n).  ∀  –D and –D is a convex cone, it follows that D – D = – n > N , k > K (n).  ∀  D – D = – G, we obtain z k k G(x ) + D. Hence, there exists z ¯ ∈ n n k k Eˆ . It follows from (.) that t (x – x) ˆ → n n → ∞ , and hence, u n ∈ R(Eˆ , x). It follows from (.) that t (u – u) x, and hence, ˆ n n ˆ → ¯ R(R(Eˆ , x), u). By Proposition ., since y ˆ ˆ ¯ ∈ R F(x, y, u, v)(x), we conclude that ˆ ˆ ˆ ˆ ¯ l R F(x, y, u, v) R R(Eˆ , x), u ˆ ˆ ˆ ˆ ˆ ˆ l clcone clcone F(Eˆ ) + C – y – v . ˆ ˆ From (.) it follows that clcone clcone F(Eˆ ) + C – y – v ˆ ˆ ∩ intcone ε ( U + B) = .  ∅ In the similar way, we conclude that This is a contradiction to (.). The proof is completed. Corollary . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D), (uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), F is second-order lower radial composed derivable at (xˆ, yˆ) in the direction (uˆ , vˆ), and G is second-order lower radial composed derivable at (xˆ, zˆ) in the direction (uˆ , wˆ). Then there exists a number εˆ ∈ (, δ) such that Proof The proof follows directly from Theorem . and Remark .(ii). Example . Let R be the set of real numbers, X = Y = Z = R, C = D = {t : t ≥ }, B = {}. Define the set-valued maps F : X → Y and G : X → Z by F(x) = G(x) = {y : y ≥ } if x ≥ , {y : y ≥ x} otherwise. zˆ ∈ G() ∩ (–D) = {}, R epi G, (, ) = (x, y) : x ∈ R, y ≥  , Rl F(, , , )(x) = Rl G(, , , )(x) = R G(, , , )(x) = [, +∞), x ∈ R, Then, the inclusions of Theorem . and Corollaries . and . are true. Theorem . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D), (uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), F is second-order lower radial derivable at (xˆ, yˆ) in the direction (uˆ , vˆ), and G is second-order radial derivable at (xˆ, zˆ) in the direction (uˆ , wˆ ). Then there exists a number εˆ ∈ (, δ) such that RlF(xˆ, yˆ, uˆ , vˆ)(x), RG(xˆ, zˆ, uˆ , wˆ )(x) ∩ Proof On the contrary, suppose that (.) does not hold. Then, for any ε ∈ (, δ), there exist x¯ ∈ dom RlF(xˆ, yˆ, uˆ , vˆ) ∩ dom RG(xˆ, zˆ, uˆ , wˆ ), y¯ ∈ RlF(xˆ, yˆ, uˆ , vˆ)(x¯), and z¯ ∈ RG(xˆ, zˆ, uˆ , wˆ )(x¯) such that  R G(x, z, u, w)(x) it follows that ˆ ˆ ˆ ˆ ¯ Hence, there exist t > , x n x, and z z such that x + t u + ˆ n ˆ t x , z + t w + t z n ˆ n ˆ n n n ∈ n ∈ The set of positive integers is denoted by N . From (.) and z z it follows that there  ∈ N such that n ∩ x + t u + ˆ n ˆ n ∈ n ∈ D and –D are convex cones, we obtain n ∈ –D – D – D = – It follows from (.) that there exists z ˜n ∈ n ∈ {˜n} Since (.) and D is a convex cone, we obtain D – D = – x it follows that x ¯ ∈  R (Eˆ , x, u). By Proposition . and y ˆ ˆ ¯ ∈   R F(x, y, u, v) R (Eˆ , x, u) ˆ ˆ ˆ ˆ ˆ ˆ l clcone cone F(Eˆ ) + C – y – v . ˆ ˆ It follows from (.) that intcone ε ( U + B) is open, we obtain clcone cone F(Eˆ ) + C – y – v ˆ ˆ ∩ cone cone F(Eˆ ) + C – y – v ˆ ˆ ∩ cone ε ( U + B) is a pointed cone, it follows that F(Eˆ ) + C – y – v ˆ ˆ ∩ F(Eˆ ) + C – y v – ˆ ∩ ˆ In a similar way, we conclude that cone ε ( U + B) is a pointed cone, we obtain ˆ ∈ Corollary . Suppose that (x, y) is a Henig efficient element of (VP), z ˆ ˆ ˆ ∈ (–C) (–D), F is second-order lower radial derivable at (x, y) in the direction × ˆ ˆ there exists a number R F(x, y, u, v)(x), R G(x, z, u, w)(x) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ l l D) = for all x Proof The proof follows immediately from Theorem . and Remark .(ii). Corollary . Suppose that (xˆ, yˆ) is a Henig efficient element of (VP), zˆ ∈ G(xˆ) ∩ (–D), (uˆ , vˆ, wˆ ) ∈ X × (–C) × (–D), B is a base of C, F is second-order lower radial derivable at (xˆ, yˆ) in the direction (uˆ , vˆ), and G is second-order lower radial derivable at (xˆ, zˆ) in the direction (uˆ , wˆ). Then there exists a number εˆ ∈ (, δ) such that Proof It is similar to the proof of Corollary .. Example . Let R be the set of real numbers, X = Y = Z = R, C = D = {t : t ≥ }, and B = {}. Define the set-valued maps F : X → Y and G : X → Z by F(x) = {y : y ≥ }, x ∈ R, G(x) = {y : y ≥ x}, x ∈ R. zˆ ∈ G() ∩ (–D) = {}, Then, the inclusions of Theorem . and Corollaries . and . are true. min f (x), s.t. g(x) ∈ –D, x ∈ X. Similarly to Definition . in [], we introduce the following second-order generalized lower (upper) directional derivative for vector-valued functions. Remark . When the set-valued map F becomes to a vector-valued function f , which is Fréchet differentiable at xˆ, letting vˆ := f (xˆ)uˆ , we have Remark . When the set-valued map F becomes to a vector-valued function f , which is Fréchet differentiable at xˆ, letting vˆ := f (xˆ)uˆ , we have Corollary . Suppose that (xˆ, yˆ) is a Henig efficient element of (P) and g(xˆ) ∈ –D. Then there exists a number εˆ ∈ (, δ) such that D˜lf+(xˆ, uˆ )(x), D˜g+(xˆ, uˆ )(x) ∩ 5 Conclusions In this paper, we introduced some new kinds of lower radial tangent cone, second-order lower radial tangent set, and second-order radial tangent set. By virtue of these concepts, second-order radial composed tangent derivative, second-order radial tangent derivative, second-order lower radial composed tangent derivative, and second-order lower radial tangent derivative for a set-valued map are introduced. Compared with the secondorder composed contingent derivative D F(xˆ, yˆ, uˆ , vˆ) introduced in [, ], the secondorder contingent derivative DF(xˆ, yˆ, uˆ , vˆ), second-order radial composed tangent derivative R F(xˆ, yˆ, uˆ , vˆ), and second-order radial tangent derivative RF(xˆ, yˆ, uˆ , vˆ), second-order lower radial composed tangent derivative Rl F(xˆ, yˆ, uˆ , vˆ), and second-order lower radial tangent derivative RlF(xˆ, yˆ, uˆ , vˆ) have nice properties: Rl F(xˆ, yˆ, uˆ , vˆ) R R(E, xˆ), uˆ ⊂ clcone clcone F(E) + C – yˆ – vˆ derivatives of the objective function and constraint function are separated. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed to each part of this work equally, and they all read and approved the final manuscript. Authors’ information Yihong Xu (1969-), Professor, Doctor, the major field of interest is the set-valued optimization. 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Bihang Xu, Zhenhua Peng, Yihong Xu. Second-order lower radial tangent derivatives and applications to set-valued optimization, Journal of Inequalities and Applications, 2017, 7, DOI: 10.1186/s13660-016-1275-x